These quantities are defined in, for example, Harvey (1989)
section 5.4. In fact, there he defines the standardized innovations in
equation 5.4.1, but in his version they have non-unit variance, whereas
the standardized forecast errors computed by the Kalman filter here
assume unit variance. To convert to Harvey’s definition, we need to
multiply by the standard deviation.

Harvey notes that in smaller samples, “although the second moment
of the \(\tilde \sigma_*^{-1} \tilde v_t\)’s is unity, the
variance is not necessarily equal to unity as the mean need not be
equal to zero”, and he defines an alternative version (which are
not provided here).